This paper investigates the properties of Quasi Maximum Likelihood estimation of an approximate factor model for an $n$-dimensional vector of stationary time series. We prove that the factor loadings estimated by Quasi Maximum Likelihood are asymptotically equivalent, as $n\to\infty$, to those estimated via Principal Components. Both estimators are, in turn, also asymptotically equivalent, as $n\to\infty$, to the unfeasible Ordinary Least Squares estimator we would have if the factors were observed. We also show that the usual sandwich form of the asymptotic covariance matrix of the Quasi Maximum Likelihood estimator is asymptotically equivalent to the simpler asymptotic covariance matrix of the unfeasible Ordinary Least Squares. These results hold in the general case in which the idiosyncratic components are cross-sectionally heteroskedastic, as well as serially and cross-sectionally weakly correlated. This paper provides a simple solution to computing the Quasi Maximum Likelihood estimator and its asymptotic confidence intervals without the need of running any iterated algorithm, whose convergence properties are unclear, and estimating the Hessian and Fisher information matrices, whose expressions are very complex.

翻译：本文研究了$n$维平稳时间序列向量在近似因子模型下拟极大似然估计的性质。我们证明，当$n\to\infty$时，通过拟极大似然估计得到的因子载荷与主成分估计的因子载荷渐近等价。这两种估计量还进一步与因子可观测时的不可行普通最小二乘估计量渐近等价（当$n\to\infty$时）。同时，我们表明拟极大似然估计量渐近协方差矩阵的常见三明治形式与不可行普通最小二乘估计的简化渐近协方差矩阵渐近等价。这些结论在更一般的情形下成立，即异质成分存在截面异方差性，同时存在序列和截面弱相关性。本文为计算拟极大似然估计量及其渐近置信区间提供了一个简单方案，无需运行任何收敛性不明确的迭代算法，也无需估计表达式极为复杂的海森矩阵和费希尔信息矩阵。